The handle http://hdl.handle.net/1887/58471 holds various files of this Leiden University dissertation.
Author: Buters, F.M.
Title: Where photons meet phonons
Issue Date: 2017-12-21
3
Experimental exploration of the optomechanical attractor diagram and its dynamics
In this chapter we investigate the optomecanical interaction in more detail. In par- ticular we explain how the motion of the mirror leads to phase modulation of the cavity field, creating optical side-bands. These side-bands interact back again with the mirror, leading to intricate dynamics.
We demonstrate this experimentally by exploring the attractor diagram of an op- tomechanical system where the optical forces compensate for the mechanical losses.
In this case stable self-induced oscillations occur but only for specific mirror am- plitudes and laser detunings. We demonstrate that we can amplify the mechanical mode to an amplitude 500 times larger than the thermal amplitude at 300 K. The lack of unstable or chaotic motion allows us to manipulate our system into a nontrivial steady state and explore the dynamics of self-induced oscillations in great detail.
The work presented in this chapter is based on: F.M. Buters, H.J. Eerkens, K. Heeck, M.J. Weaver, B. Pepper, S. de man and D. Bouwmeester, ”Experimental exploration of the optomechanical attractor diagram and its dynamics”, Phys. Rev. A 92, 013811 (2015) and F.M. Buters, H.J. Eerkens, K. Heeck, M.J. Weaver, B. Pepper, P. Sonin, S. de man and D. Bouwmeester, ”Large parametric amplification in an optomechanical system”, Physica Scripta 2015.T165 (2015)
3.1 Introduction
Laser or microwave cooling of a mechanical degree of freedom has led several groups to come close to or even reach the quantum-mechanical ground state of a macro- scopic harmonic oscillator [39, 40, 41]. This has opened up many new research av- enues to investigate the foundations of quantum mechanics [42], novel decoherence mechanisms [9, 14, 43], and strong photon-phonon coupling [44, 45, 46]. Besides cooling, heating of the mechanical degree of freedom is possible, leading to para- metric instabilities, self-induced oscillations, and even chaos. Braginsky et al. have derived the condition for achieving parametric instability in a Fabry-Perot interfer- ometer such as LIGO [47], which is still a topic of interest [48]. The theoretical frame- work has been expanded by Marquardt et al., with the introduction of an attractor diagram and an expression for the optomechanical gain [49, 50]. From an experi- mental point of view Carmon et al. showed how self-induced oscillations of the me- chanical mode are imprinted on the cavity output field [51, 52]. Finally the transition from self-induced oscillation to chaos has been investigated with some interesting prospects for observing the quantum to classical transition [53, 54].
The dynamics of self-induced oscillations are best understood using an attractor diagram. So far only a small part of this diagram has been explored experimentally by Metzger et al. with the photothermal effect as the driving force [55]. Little effort has been made to investigate the attractor diagram experimentally using radiation pressure force. It has therefore been to date an open problem to explore the full at- tractor diagram experimentally [16]. One reason for this is that a transition from self- induced oscillations to chaotic mirror motion can occur due to second-order effects, such as absorption-induced heating of the optical components [51, 53]. This restricts the exploration of the attractor diagram to small values of the mirror amplitude.
Here we demonstrate an optomechanical setup, consisting of a Fabry-Perot cav- ity with a trampoline resonator, that does not suffer from optical absorption in the mirrors. Not only does this enables us to explore a large part of the attractor diagram in a controlled fashion, but also we find surprisingly rich dynamics and non-trivial steady states of our optomechanical system.
3.2 Theoretical model
Our optomechanical system is described by two coupled equations of motion:
da(t)
dt = −κ
2 a(t) + i(∆ + Gx(t))a(t) +√
κexsin (3.1) d2x(t)
dt2 = −Ω2mx(t) − Γm
dx(t) dt +~G
m |a(t)|2, (3.2)
in which a is the cavity field and x is the mirror displacement. The parameters in Eqs. (3.1) and (3.2) are defined as follows: sinis the laser field, κ is the overall cavity decay rate, κex is the cavity entrance loss rate, ∆ = ωL− ωcav is the laser detuning defined as the difference between cavity and laser frequencies, the optical frequency shift per displacement G = ωcav/L, with L being the length of the cavity, Ωm as
3.2 Theoretical model 23
the fundamental mode frequency of the mechanical oscillator, Γmas the mechanical damping rate and m as the mode mass of the harmonic oscillator. Thermal and mechanical noise sources have been neglected; an important assumption that will be justified for our optomechanical system by the results below.
The optomechanical attractor diagram displays the optomechanical gain ζopt, the ratio of the radiative force Pradand frictional losses Pf ric, as a function of laser de- tuning ∆ and mirror amplitude A. From Eqs. (3.1) and (3.2) an expression for ζopt
can be derived [49],
ζopt(∆, A) = Prad
Pf ric = − 1 Γm
2~Gκexs2in mΩmA Im(X
n
a∗n+1an) (3.3)
with
an = Jn(−GA/Ωm)
κ/2 − i ˜∆ + inΩm, (3.4)
in which αn is the nth harmonic (or sideband) in the optical field created by the mirror motion, Jnis the Bessel function of the first kind, and ˜∆is the effective laser detuning defined as ˜∆ = ωL− ωcav+ G¯x, where ¯xis the static displacement of the mirror due to the radiation pressure. For most situations, including ours, the static displacement is negligible and ∆ ≈ ˜∆.Stable self-induced oscillations occur when ζopt(∆, A) = 1, whereas amplification (attenuation) of the mechanical mode occurs when ζopt(∆, A) > 1(ζopt(∆, A) < 0).
To give a better understanding of the phenomena of self-induced oscillations we describe the parametric process in an optomechanical system qualitatively and con- nect this back to Eqs. (3.1)-(3.4). When the incoming laser field at frequency ωL
interacts with the moving mirror (the term Gx(t)a(t) in Eq.(3.1)), the laser field is phase modulated such that sidebands appear at ω = ωL± nΩmwith n a positive integer. The number of sidebands that appear depends on the modulation depth φo = −GA/Ωm. The larger the amplitude, the more sidebands or harmonics of the mechanical frequency Ωmare imprinted on the cavity field, as is clear from Eq.(3.4) where φ0is the argument of the Bessel function. Note that to take into account the enhancement of the optical field by the presence of a cavity, Jn(−GA/Ωm)is scaled for the cavity line shape (see denominator of Eq.(3.4)). If the cavity field, with all the harmonics, interact again with the mirror, only resonant force components con- tribute to the motion of the mirror (resonant terms of ~Gm |a|2). These resonant force components arise by the non-linear mixing of neighboring harmonics anat the mov- ing mirror (the term P
na∗n+1an in Eq.(3.3)). By taking the imaginary part of this sum, the damping due to the optical field is obtained. Alternatively, by taking the real part the term responsible for the optical spring effect is obtained. By changing the laser frequency detuning ∆ or varying the amplitude of the mirror A, the optical damping and thereby also the optomechanical gain ζoptis either positive (attenua- tion) or negative (amplification).
One way to map out the attractor diagram ζopt(∆, A)is to measure the mirror am- plitude while varying the laser detuning. Such measurement schemes have already successfully been used for demonstrating optical cooling. With optical cooling, the change in cavity resonance frequency due to the motion of the mirror is usually much
Nd:YAG
1064 nm OI EOM
Vacuum chamber+ +
R T
300 μm
trampoline resonator
Figure 3.1: Schematic of the set up. A piezo tunable cw Nd:YAG laser is passed through an optical isolator (OI) and a 9.5 MHz electro-optical modulator (EOM) before it enters a fiber circulator that is fed into a vacuum chamber that contains a 5 cm Fabry-Perot cavity. Both the transmitted and the reflected intensities are recorded with photo-detectors. The inset shows an optical image of the trampoline resonator.
smaller than the linewidth of the cavity resonance, i.e., GA κ. In the optical field only the first sideband is visible, and the magnitude of this sideband is linear with mirror amplitude. For optical excitation, however, the change in cavity resonance frequency can be much larger than the cavity linewidth, i.e., GA κ, resulting in multiple sidebands present in the optical field. The linear relation between the first sideband and the mirror amplitude no longer holds. Now the mirror amplitude can only be obtained by taking into account all optical sidebands.
3.3 Experimental setup
To map out the attractor diagram we make use of a 5 cm long Fabry-Perot cavity operating around 1064 nm, with a trampoline resonator as one of the end mirrors [33]. By using a multilayer Bragg stack on both cavity mirrors, absorption losses are minimized to about 1 ppm. The system is placed inside a vacuum chamber with a vibration isolation system containing several eddy-current dampers. All measure- ments are performed at room temperature. A schematic of the set-up is given in Fig. 3.1. We use a piezo tunable cw Nd:YAG laser and control it with a typical scan speed of dωdtL = 100 − 400MHz/s, which is slow compared to the cavity build up time, i.e. dωdtL κ/τ with κ as the cavity linewidth and τ as the cavity lifetime. An EOM at 9.5 MHz is used to calibrate the laser detuning. The mechanical properties of the trampoline resonator are characterized by measuring the thermal noise spec- trum and the optical properties by performing an optical ring-down measurement [33]. Both transmitted and reflected cavity light are detected using photo-detectors, and the data-acquisition is performed using a digital storage scope. For our system only the fundamental mechanical mode and fundamental optical mode (TEM00) are relevant. The parameters for our system are the following: κ = 175 × 103× 2π rad/s, κex = 50 × 103× 2π rad/s, Ωm = 343 × 103× 2π rad/s, Γm = 1.7 × 2πrad/s at a
3.4 Results 25
pressure of 10−6 mbar and m = 110 × 10−12 kg. To achieve a sufficiently large op- tomechanical gain, the input laser power should also be sufficiently large. A typical laser input power of 50 to 100 µW is used, corresponding to an intracavity photon number of 2.8 − 5.6 × 108.
3.4 Results
Figure 3.2(a) shows the optical intensity transmitted by the cavity when the laser is scanned back and forth across the cavity resonance. Several peaks are visible, not only at the cavity resonance ∆/Ωm = 0, but also at multiples of ∆/Ωm. In the ap- pendix of this chapter, the behavior of these peaks is investigated in more detail. The appearance of sidebands can be explained as follows. Suppose the laser frequency is at ωL = ωcav+ Ωmand the amplitude of the mirror is small such that only the first sideband is created by the moving mirror at frequencies ω = ωL± Ωm. Only the Stokes sideband at ω = ωcavis resonant with the cavity and enhanced, whereas the anti-Stokes sideband at ω = ωcav+ 2Ωmis suppressed. So the interaction of the blue detuned laser field with the resonator leads to a resonant field in the cavity. The non-linear interaction of the resonant cavity field plus the incoming laser field with the mirror lead to a resonant driving force. By creating sidebands, the mirror gen- erates its own driving force, which increases the mirror amplitude. The increased mirror amplitude leads to a stronger modulation of the cavity field, and this process repeats until the gain is reduced to ζopt = 1(limit cycle behavior). When the laser detuning is slowly increased further, the process repeats whereby the ever increasing mechanical motion allows sideband generation to drive the mirror to larger ampli- tudes. This process continues until the laser is swept back rapidly to ∆/Ωm= −30. At first the laser detuning and mirror amplitude do not match to produce an optical force that influences the mirror motion. The mirror amplitude decreases only due to the intrinsic mechanical damping. Although the laser detuning is slowly increased towards zero detuning at some point, in this case at ∆/Ωm= −12, the laser detuning and mirror amplitude are such that sidebands created by the mirror motion result in an optical force. However the sign of the optical force has changed compared to the situation with positive detuning. Instead of parametric amplification, now paramet- ric attenuation occurs, resulting in a decrease in mirror amplitude. The interaction of the laser field with the resonator again leads to a resonant cavity field, result- ing in peaks at multiples of ∆/Ωm also for negative laser detunings. This is only visible when the mirror amplitude was driven to large values previously. Driven os- cillations at negative laser detunings reveal therefore something about the state and history of the system and are non-trivial.
To compare the experimental result of Fig. 3.2(a) with theory, a numerical sim- ulation is performed with the same experimental parameters. For this we solve numerically Eqs. (3.1) and (3.2) using the following initial conditions: α(0) = 0, α0(0) = 0, x(0) = x0and x0(0) = 0where x0denotes the initial mirror amplitude x0. The value for x0 is chosen to correspond to the thermal mirror amplitude at 300K:
x0 =pkbT /mΩ2m. Although no mechanical and thermal noises are required to re- produce the experimental results, an initial mirror amplitude is needed to start the
20 15 10 5 0 5 10 Laser detuning (units of
∆/Ωm) 0
100 200 300 400 500 600 700 800
Mirror amplitude (pm)
(c)
ζ
opt ¿1 1 ζ
opt À1
0 50 100 150 200
Time (ms) 0.0
0.2 0.4 0.6 0.8 1.0
Transm. intensity (arb. units)
30
20 10 0 10
Laser detuning (units of ∆/Ωm)
(b) Simulation
0 50 100 150 200
Time (ms) 0.0
0.2 0.4 0.6 0.8 1.0
Transm. intensity (arb. units)
30
20 10 0 10
Laser detuning (units of ∆/Ωm)
(a) Experiment
Figure 3.2:(Color) A closed cycle across the attractor diagram. (a) Intensity transmitted by the cavity for two consecutive periods of a controlled laser detuning sweep. (b) Simulation based on Eqs. (3.1)-(3.2). (c) Attractor diagram corresponding to our experimental parameters. The path followed in the experiment is indicated by the arrows.
3.4 Results 27
parametric process.
The results of the simulation, depicted in Fig. 3.2(b), are in good agreement with the experimental results of Fig. 3.2(a). This indicates that our earlier assumption not to include thermal and mechanical noise in Eqs. (3.1) and (3.2) is justified. Fur- thermore, we do not need to include any second order effects such as heating of the mirror substrates due to absorption. Although from the experimental data the mirror amplitude is not obtained directly, the numerical simulations do contain the mirror amplitude. By plotting the attractor diagram according to Eq. (7.4) together with the mirror amplitude obtained from the simulations, we can visualize the traversed path across the attractor diagram.
In Fig. 3.2(c) the attractor diagram is displayed together with the evolution of the mirror amplitude (indicated by the arrows). The amplitude follows a deterministic path through the diagram. Along this path the optomechanical gain varies. When the gain is large, the path closely follows the ζopt= 1contour, whereas in the regions with moderate gain the changing laser detuning prevents the mirror amplitude from reaching the ζopt = 1contour as closely. Specifically, for positive laser detunings ζopt≥ 1 and for negative laser detunings ζopt≤ 1. It is also worthwhile to emphasize that the mirror amplitude changes on the time scale of the laser scan speed, much slower than the oscillation frequency of the mirror or the cavity lifetime. So far we have thus been discussing the dynamics of a driven, quasi-static, system. However, also interesting dynamics occur on the time scale of the mechanical resonator.
Theoretically the increase in the mirror amplitude, as shown in Fig. 3.2(c), should be visible as an increase in the number of harmonics present in the output field [49].
This is verified by analyzing the fast modulation present in the reflected intensity for several different detunings corresponding to the white dots in Fig. 3.2(c). We have analyzed the reflected intensity as it is picked up by a faster photodetector in our experimental configuration. However the same features should also be visible in the transmitted intensity.
In Fig. 3.3 we compare experimental and numerical results for these fast modu- lations. For clarity an offset is removed, and the figures are rescaled. The excellent agreement between theory and experiment confirms once more that we have ex- plored in detail the boundary (lowest branch where ζopt= 1) of the attractor diagram and that this method is suited for further exploration of the attractor diagram. Fur- thermore, we have significantly amplified the motion of our mechanical resonator, using large intracavity power, without any sign of unstable or chaotic behavior.
To demonstrate that we have full control over our system, we change the starting conditions of our laser frequency sweep after performing a cycle similar to the one displayed in Fig. 3.2. When the mirror amplitude is large, changing the laser detun- ing slightly makes it possible to skip from the boundary branch to another branch.
In this way different branches in the attractor diagram can be explored.
Figure 3.4(a) shows the results of two cycles across the attractor diagram along a different branch. The scale for the transmitted intensity is the same as in Fig. 3.2(a).
Although the experimental conditions have only changed a little, the result is quite different from Fig. 3.2(a). Still multiple peaks at ∆/Ωmare visible, but the main cavity resonance at ∆/Ωm= 0is reduced significantly compared to these sidebands.
Also a distinctive dip is visible, indicated with ”*”. To verify that the features of Fig.
0.0 0.2 0.40.6 0.81.0
Refl. intensity (arb. units)
Experiment
(a)
∆/Ωm=1 Simulation
0.0 0.2 0.40.6 0.81.0
Refl. intensity (arb. units)
(b)
∆/Ωm=30.0 0.20.4 0.6 0.81.0
Refl. intensity (arb. units)
(c)
∆/Ωm=50.0 0.5 1.0 1.5 2.0
Time (units of Ωmt/2π) 0.0
0.20.4 0.6 0.81.0
Refl. intensity (arb. units)
(d)
0.0 0.5 1.0 1.5 2.0
Time (units of Ωmt/2π)
∆/Ωm=10
Figure 3.3: Detailed time traces of the reflected cavity intensity for different laser detunings.
The left column shows the measurements and the right column shows numerical solutions to Eqs. (3.1) and (3.2). (a)-(d) correspond to specific detunings indicated with white dots in Fig.
3.2(c).
3.4 Results 29
10 5 0 5 10 15
Laser detuning (units of
∆/Ωm) 0
200 400 600 800 1000 1200 1400 1600
Mirror amplitude (pm)
(c)
*
ζ
opt ¿1 1 ζ
opt À1
2 1 0 1 2 3 4 450 500
550 600
0 20 40 60 80 100 120 140 160 Time (ms)
0.00 0.05 0.10 0.15 0.20
Transm. intensity (arb. units)
30
20 10 0 10 20
Laser detuning (units of ∆/Ωm)
(b)
* *
Simulation
0 20 40 60 80 100 120 140 160 Time (ms)
0.00 0.05 0.10 0.15 0.20
Transm. intensity (arb. units)
30
20 10 0 10 20
Laser detuning (units of ∆/Ωm)
(a)
* *
Experiment
Figure 3.4:(Color) Exploring a higher branch in the attractor diagram. (a) Intensity transmit- ted by the cavity for two consecutive periods of a controlled laser detuning sweep. The scale for the transmitted intensity is the same as in Fig. 3.2(a) (b) Simulation based on Eqs. (3.1) and (3.2). (c) Attractor diagram together with the path followed in the experiment. Before switching to a higher branch, the system is initialized (dashed line) using a similar detuning sweep as in Fig. 3.2(c). Inset: Zoom of region around zero detuning.
3.4(a) are captured by the theoretical model of Eqs. (3.1)-(3.2), a numerical simulation is performed with the same experimental parameters. The qualitative agreement be- tween experiment and simulation shows that the model is still valid for our system.
Furthermore, from the simulation we can again extract the mirror amplitude and use this together with the attractor diagram to explain the features of Fig. 3.4(a).
Figure 3.4(c) shows this attractor diagram. The black dashed line shows the ini- tialization, which is similar to the cycle performed in Fig. 3.2, but now the laser detuning is set back to just ∆/Ωm = −5to reach a different branch. Note that the initialization is not shown in Figs. 3.4(a) and 3.4(b). The solid black line shows the evolution of the mirror amplitude during one cycle. The largest mirror amplitude achieved in this experiment is roughly 1600 pm, more than 500 times the amplitude at 300K without any sign of chaotic or unstable behavior.
For the steady state cycles of Fig. 3.4(a) the reduction in the transmitted inten- sity at the cavity resonance (∆/Ωm= 0) is now readily explained: The large mirror amplitude reduces the time the cavity is resonant with the input field, therefore less intracavity field is built-up, resulting in a reduction in transmitted intensity.
The inset of Fig. 3.4(c) shows the region where a change from one stable branch to another occurs. This transition occurs at {∆/Ωm= 1.5, A ≈ 510 pm}. At this point the mirror amplitude stays constant along a contour where ζopt = 1. This point coincides with the distinctive dip in Fig. 3.4(a). When the mirror amplitude does not change, no optical driving force occurs and no sideband is visible in the opti- cal output. Even more interesting is the surrounding area of the attractor diagram.
At {∆/Ωm= 1.5, A ≈ 510 pm}any small change in mirror amplitude is significantly amplified: if the mirror amplitude increases slightly, ζopt 1 and the mirror ampli- tude will increase significantly. Similarly, if the mirror amplitude decreases slightly, ζopt 1 and the mirror amplitude will decrease significantly. The inset therefore highlights a bistability: A small perturbation of the mirror motion will result in a large change in the mirror amplitude. However, our results show that in a clean sys- tem such as ours, we can ”walk” through such unstable regions on a well-defined path.
3.5 Conclusion
With the absence of any chaotic or unstable behavior our optomechanical system is only described by two equations (Eqs. (3.1) and (3.2)). This has allowed us to explore in detail the optomechanical attractor diagram and the dynamics of self-induced oscillations. By performing a laser frequency sweep, multiple stable branches in the attractor diagram are explored. Furthermore, we have demonstrated non-trivial dynamics, such as driven oscillations for negative laser detunings and the presence of a bistability.
Threshold behavior of self-induced oscillations
In early stages of this work, the threshold behavior of self-induced oscillations was investigated. In particular, the influence of laser input power and mechanical decay
3.5 Conclusion 31
Below threshold Above threshold
Experiment HaL
0 4 8 12 16 20
0.0 0.2 0.4 0.6 0.8 1.0
0 3 6 9 12 15
Laser detuningDWm
TransmittedintensityHa.u.L
TimeHmsL
Figure 3.5: Resulting transmitted intensity for a linear laser frequency sweep at 400 MHz/s.
Additional side-band are only visible with sufficient laser power.
æ æ æ æ
æ æ
æ æ
æ æ HaL æ
1 2 3 4 5 6
0 5 10 15 20 25 30
Laser input powerHa.u.L PlateaulengthHDWmL
æ æ æ
æ Fit with 1
Gm
HbL
0 50 100 150 200 250 300
0 5 10 15 20 25 30
Gm2ΠHHzL
PlateaulengthHDWmL
Figure 3.6: Verification of optomechanical gain. (a) Influence of laser input power on plateau length for fixed mechanical damping rate Γm (Qm = 34000).(b) Influence of mechanical damping rate Γmon plateau length for fixed laser input power.
rate Γm. A set-up with the following parameters was used: κ = 300 × 103× 2π rad/s, κex = 50 × 103× 2π rad/s, Ωm = 300 × 103× 2π rad/s, Γm = 8.8 × 2π rad/s (or Qm= 34000) at 10−6mbar and m = 110 × 10−12kg.
In Fig. 3.5 the transmitted intensity is shown when a linear laser frequency sweep is performed for low (red) and high (blue) laser power. Side-bands are only visible above a minimum laser power and the length of this plateau of side-bands is a mea- sure for the condition ζopt(∆, A) > 1. This length can be used to show the threshold behavior, as is shown in Fig. 3.6(a). Only with a certain minimal laser power do the self-induced oscillations start. In accordance with Eq. (3.3), the number of side- bands visible scales linearly with power.
Besides laser power, the influence of the mechanical damping rate Γmwas also investigated by changing the pressure in the vacuum chamber and thereby altering Γm. The results, shown in Fig. 3.6(b), demonstrate the correct 1/Γmdependency as predicted by Eq. (3.3). Therefore, the number of side-bands visible in Fig. 3.5 are a correct measure to investigate the condition ζopt(∆, A) > 1in an optomechanical system.
Acknowledgments
The authors would like to thank H. van der Meer for technical assistance and sup- port. The authors are also grateful for useful discussions with M.P. van Exter, W.
Loeffler, and G. Welker. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM) and of the NWO VICI research pro- gram, which are both part of the Netherlands Organisation for Scientific Research (NWO). This work is also supported by National Science Foundation Grant No.
PHY-1212483.
